NAIROBI (Reuters) -Kenyan police fired tear gas to disperse protesters in the capital Nairobi on Tuesday as a nationwide demonstration against proposed tax hikes got underway.

Organisers have called for protests and a general strike against the tax legislation, seeking to build on momentum that has, in the span of a week, turned an online, youth-led movement into a major headache for the government.

The finance bill aims to raise an additional $2.7 billion in taxes as part of an effort to tame a heavy debt load, with interest payments alone consuming 37% of annual revenue.

President William Ruto won an election almost two years ago on a platform of championing Kenya's working poor, but has been caught between the competing demands of lenders like the International Monetary Fund, which is urging the government to cut deficits, and a population reeling from cost-of-living increases.

Thousands took to the streets of Nairobi and several other cities during two days of protests last week.

Police fired tear gas as small groups of protesters began to gather in the Central Business District and the informal settlement of Kibera early on Tuesday, Reuters journalists said. Some of the protesters were chanting anti-Ruto slogans.

Hundreds of protesters were also marching through the streets of the coastal city of Mombasa, Kenyan television showed.

Although the protesters last week in Nairobi were almost entirely peaceful, according to Reuters reporters and human rights organisations, police repeatedly fired tear gas and water cannon. Two people were killed, with one struck by a gunshot and another by a tear gas cannister, human rights groups said.

The main association of Kenyan lawyers also said in a statement on Monday that police had unlawfully detained and held incommunicado at least five people involved in the protests over the last 72 hours.

A police spokesperson did not respond to requests for comment. In remarks on Sunday, Ruto praised the protesters, saying they had been peaceful and that the government would engage with them on the way forward.

While protesters initially focused on the finance bill, their demands have broadened, with many chanting last week: "Ruto must go!"

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Political analysts say the protests represent a particular challenge for Ruto because, unlike previous demonstrations led by political parties, they lack an official leader who can be mollified through private negotiation and inducements.

The government has already made some concessions, promising in amendments to the bill to scrap proposed new taxes on bread, cooking oil, car ownership and financial transactions. But that has not been enough to satisfy protesters, who want the entire bill scrapped.

On Tuesday afternoon, lawmakers will debate the amendments, which the finance ministry says would blow a 200 billion Kenyan shilling ($1.56 billion) hole in the 2024/25 budget, and compel the government to make spending cuts or raise taxes elsewhere.

(Reporting by Duncan Miriri and Hereward Holland; Editing by Aaron Ross, Christina Fincher and Angus MacSwan)